Blockwise Zassenhaus Conjectures (Defect 1)

• The paper establishes that in defect 1 blocks, any torsion unit of order p is rationally conjugate to ± a group element, confirming a blockwise version of the Zassenhaus conjectures.
• It employs modular representation theory through Brauer trees and detailed module filtrations to impose strict linear constraints on possible unit structures.
• Explicit constructions demonstrate local units with composite orders that challenge classical results and suggest new avenues for exploring counterexamples in finite simple groups.

The blockwise variant of the Zassenhaus conjectures frames the problem of describing torsion units in integral group rings through the lens of modular representation theory, specifically by restricting attention to units supported in a single block of defect 1. Focusing on modules and characters attached to blocks leads to significant advances in the understanding of rational conjugacy and the possible existence of counterexamples among finite simple groups. This formulation leverages the structure theory of blocks, the combinatorics of Brauer trees, and lattice-theoretic methods to decide the existence and properties of finite order units within block components, providing both global theorems and explicit constructions in specific settings (Eisele et al., 2022).

1. Formulation of the Blockwise Zassenhaus Conjecture for Defect 1

Let $G$ be a finite group and $p$ a prime. The central primitive idempotent $b \in Z(\mathbb{Z}_p G)$ defines a $p$-block $B$ of defect 1. The defect group $D$ is cyclic of order $p$: $D \cong C_p$. Associated with $B$ are the sets of ordinary and Brauer characters, denoted $\Irr(B)$ and $\IBr(B)$ respectively. For such a block, the blockwise Zassenhaus conjecture asserts:

• If $u \in V(\mathbb{Z} G)$ is a torsion unit of order $p$ and $u$ is supported in $B$ (i.e., $ub = bu$), then $u$ is conjugate in the rational group algebra $\mathbb{Q} G$ to $\pm g$ for some $g \in G$ with $gb = bg$.

This blockwise assertion reflects the conjecture’s global content for elements whose class lies in a block of cyclic defect, reducing the problem to analysis of block-local structure and modular character relationships (Eisele et al., 2022).

2. Representation-Theoretic Infrastructure: Brauer Trees and Filtrations

Each defect 1 block $B$ is parametrized through its Brauer tree—a combinatorial invariant encoding relationships between ordinary and Brauer characters and the reduction mod $p$ of irreducible lattices. The vertices are $\Irr(B)$, and edges are $\IBr(B)$. The alternation identity on $p'$-elements and the multiplicity formula of Luthar–Passi yield linear constraints on the possible distribution of partial augmentations of torsion units, which in turn restrict the possible existence of non-trivial units in the block.

Key representation-theoretic mechanisms are:

• Decomposition matrix governed by the Brauer tree.
• Filtration of modules by indecomposable summands $I_\ell$ (unique $k C_p$-modules of dimension $\ell$).
• Galois invariance of the constituents and compatibility with the structure of $\O[\eta_i]$-lattices.
• Skew-tableau combinatorics for arranging module filtrations consistent with block and lattice properties (Eisele et al., 2022).

3. Deciding Existence of Finite Order Units in Defect 1 Blocks

A method for determining the existence of units in $\mathbb{Z}_p G$ of defect 1 is established as follows: given a cyclic group $C = \langle u \rangle \cong C_n$, work over a splitting complete DVR $\O$ with residue field $k$ of characteristic $p$ containing all $n$-th roots of unity. The existence of a full $\O C$-lattice $L$ affording a prescribed module $N$—filtered so that successive quotients are direct sums of indecomposable modules $I_\ell$ according to the combinatorics of the block—corresponds bijectively to the realizability of the unit $u$ in the block.

The construction relies on:

• The explicit filtration

$$0 = N_0 \subset N_1 \subset \cdots \subset N_{e+1} = N$$

where the layers correspond to block combinatorics and Galois orbits of characters.

• Use of projective covers, the Heller operator $\Omega$, and stepwise lifting extensions.
• The ability to translate between reductions mod $\pi$ and the underlying $\O$-lattice structures to reconstruct candidate units (Eisele et al., 2022).

4. The Principal Theorem and Its Implications

A central result states: If $G$ has a Sylow $p$-subgroup of order $p$ and $u \in V(\mathbb{Z} G)$ is a unit of order $p$, then $u$ is rationally conjugate to $\pm g$ for some $g \in G$. The proof exploits:

• The restriction that only the principal $p$-block contains such units.
• The linear form of the Brauer tree for cyclic defect 1, localizing the problem to modules with explicit multiplicity and dimension.
• Contradictory inequalities bounding the number of indecomposable summands, arising from combinatorial techniques (e.g., skew-tableaux), which exclude the possibility of exotic units of order $p$ not arising from group elements or their negatives.

A plausible implication is that, within this restricted framework, the only non-trivial units of order $p$ are accounted for by group elements, precluding global counterexamples in this context for groups with cyclic Sylow $p$-subgroups (Eisele et al., 2022).

5. Explicit Constructions: Composite Order Units and Local-Global Phenomena

Explicit construction is achieved for a unit of order 15 in the group ring $V(\mathbb{Z}_{(3,5)}\,\mathrm{PSL}(2,16))$:

• $u \in V(\mathbb{Z}_{(3,5)}\,G)$ of order $15$ with partial augmentations $\varepsilon_{15c}(u) = -1$, $\varepsilon_{15d}(u) = 2$, where $15c$ and $15d$ are the two relevant $15$-element classes in $G$.
• Rational conjugacy classes for the powers: $u^5 \sim 3a$, $u^3 \sim 5a$.
• Verification through eigenvalue multiplicities and the HeLP package.
• Local compatibility across $p$-blocks ($p=3,5$) with their Brauer tree structures, Galois invariance, and explicit module-theoretic filtrations matching block-theoretic constraints.

This construction provides a $3$- and $5$-local counterexample to the Zassenhaus conjecture, though it does not yield a global counterexample in $\mathbb{Z} G$ due to obstructive behavior at 2-localization. The method demonstrates that blockwise filtration criteria permit the creation of local units of composite order, which elude detection by classical rational conjugacy strategies and suggest a refined perspective on the reach and limitations of Zassenhaus-type conjectures in the modular context (Eisele et al., 2022).

6. Broader Significance and Potential for Future Counterexamples

The blockwise approach not only resolves torsion unit questions for blocks of defect 1 and prime-power orders but also provides machinery to construct local units of composite order that are compatible across multiple localizations. The existence of such units raises the possibility that global counterexamples to the classical Zassenhaus conjecture may be found among simple groups, particularly via analysis of their blockwise unit structures. These developments underscore the centrality of modular representation theory and block combinatorics in contemporary investigations of group ring unit structures (Eisele et al., 2022).